A Uniform Upper Bound on the Length of MDS Codes from Genus-Two Curves

Haojie Chen

Home/Archive/arXiv

arXiv · 2610.39364

A Uniform Upper Bound on the Length of MDS Codes from Genus-Two Curves

Haojie Chen

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the maximal length of MDS algebraic geometry codes from genus-two curves over finite fields. Let $C_\mathcal{ L}(D,G)$ be such a code over $\mathbb F_q$, with length $n$ and dimension $k$. Assume that $3\leq k\leq n-3$. We prove that $n<(2q+10\sqrt q+16)/3$. This bound holds for every prime power $q$ and for both parities of $k$. It gives a uniform upper bound of $2q/3+O(\sqrt q)$ for the maximal length. The explicit bound is strictly smaller than $q+1$ when $q\geq125$.

Explore related subjects

cs.IT math.AG math.IT math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Haojie Chen. 2026-10-29. A Uniform Upper Bound on the Length of MDS Codes from Genus-Two Curves. https://arxiv.org/abs/2610.39364

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantum XYZ Stabilizer Codes

Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define $X$ and $Z$-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with $X$-, $Y$-, and $Z$-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the $Y$-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ$^2$ hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.

cs.IT↗

arXiv2026

Coded Computing for Dynamic System via Cartesian Products

Coded distributed computing (CDC) mitigates the Shuffle communication bottleneck through coded multicast, but existing schemes with predetermined worker populations do not directly adapt to worker arrivals and disconnections. To address these dynamics, we propose a dynamic CDC scheme that preserves surviving workers' caches and Reduce assignments while exploiting the storage brought by arriving clusters. The model accommodates general MapReduce tasks under a placement that is partly fixed and partly new. The scheme uses Cartesian products of cluster-wise placement delivery arrays (PDAs) for file placement. A delivery-aware greedy algorithm reassigns abandoned Reduce functions, and a Communication Array allows workers with no Reduce function to serve as coded transmitters. For arbitrary feasible disconnections, the scheme's exact communication load is derived. A converse bound under one-shot communication shows that, in the absence of disconnections, the scheme lies within a factor of 2 of the lower bound under these assumptions when new clusters arrive, and within a factor of 4 otherwise.

cs.IT↗

arXiv2026

Optimal Universal Coding of Integers

Universal coding of integers (UCI) provides prefix codes for positive integers such that, for every nonincreasing source distribution $P$, the average codeword length under $P$ stays within $K$ times $\max\{1,H(P)\}$. For a given UCI $\mathcal{C}$, the smallest such constant $K$ is called the minimum expansion factor of $\mathcal{C}$, denoted $C_{\mathcal{C}}^{*}$. Let $C^{*}\triangleq \inf \{C_{\mathcal{C}}^{*}\}$ over all UCIs. A UCI attaining $C^{*}$, or a family of UCIs attaining $C^{*}$ in the limit, is called optimal. It is currently known that $2\le C^{*}\le 2.0386$. In this paper, we construct a family of one-point plus uniform-tail distributions and prove that, for every UCI, a distribution in this family attains the worst-case ratio; thus, the family is least favorable for the UCI problem. We further establish an inequality, called the \emph{UCI inequality}, which plays the same role for UCI as the Kraft inequality does for prefix codes: for any real number $B\ge 1$, it decides whether $B$ lies below or above $C^*$. The UCI inequality gives an equivalent definition of $C^*$. By numerical computation, we determine $C^* = 2.00012475703610197861701176297579886254\cdots$, the first $38$ decimal digits being certified. Once $C^*$ is known, we can theoretically construct the optimal UCI.

cs.IT↗